Factoring Trinomials
Sometimes we must use several methods to completely factor a
polynomial.
Procedure —
General Strategy for Factoring a Polynomial
Step 1 Factor out the GCF of the terms of the polynomial.
Step 2 Count the number of terms and look for factoring patterns.
Two terms: Look for the difference of two squares, the sum of
two cubes, or the difference of two cubes.
Three terms:
• Try to factor using the patterns for a perfect
square trinomial.
• If the trinomial has the form x2 + bx + c, try to
find two integers whose product is c and whose
sum is b.
• If the trinomial has the form ax2 + bx + c, try
to find two integers whose product is ac and
whose sum is b.
Four terms: Try factoring by grouping.
Step 3 Factor completely.
To check the factorization, multiply the factors.
Example 1
Factor: 108wx2 - 36wxy + 3wy2
Solution
| Step 1 |
Factor out the GCF. Factor each term.
The GCF is 3w.
Factor out 3w. |
108wx2 =
2 · 2 · 3 · 3 · 3 · w · x · x
= 2 · 2 · 3 · 3 · 3 · w · x
· x
= 3w(36x2 - 12xy + y2) |
- 36wxy - 2 ·
2 · 3 · 3 · w · x · y
- 2 · 2 · 3 · 3 · w · x · y
|
+ 3wy2 + 3
· w · y · y
+ 3 · w · y · y |
| Step 2 |
Count the number of terms and
look for factoring patterns.
There are three terms in 36x2 - 12xy + y2.
The first term, 36x2, is a perfect square, (6x)2.
The third term, y2, is a perfect square, (y)2.
The middle term, 12xy, is twice the product of the squared terms:
12xy = 2(6x)(y)
This matches the pattern for a perfect square trinomial.
Substitute 6x for a and y for b: a2 2ab b2
`(a b)2
36x2 - 12xy + y2 = (6x)2 - 2(6x)(y) +
(y)2 = (6x - y)2 |
| Step 3 |
Factor completely.
(6x - y)2 cannot be factored further. |
Thus, the factorization is 3w(6x - y)2.
You can multiply to check the factorization.
Note:
Don’t forget the GCF, 3w, that we
factored out in Step 1.
Some polynomials cannot be factored using integers.
Example 2
Factor: x2 + 3x + 5
Solution
| Step 1 |
Factor out the GCF.
There are no factors common to all three terms, other than 1 or 1.
|
| Step 2 |
Count the number of terms and look for factoring patterns. There are three terms in x2
+ 3x + 5.
This trinomial has the form ax2 + bx + c.
To factor this trinomial we must find two integers whose product is
5 and whose sum is 3. Here are the possibilities:
| Product 1
· 5
-1 · -5 |
Sum 6
-6 |
There are no integers whose product is 1 and whose sum is 3.
So, the trinomial x2 + 3x + 5 cannot be factored using integers.
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